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arXiv:1502.06674

A Monte Carlo wavefunction method for semiclassical simulations of spin-position entanglement

Billington, C. J., C. J. Watkins, R. P. Anderson, Lisa Turner

Published Feb 24, 2015• 0 citations
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Abstract

We present a Monte Carlo wavefunction method for semiclassically modeling spin-$\frac12$ systems in a magnetic field gradient in one dimension. Our model resolves the conflict of determining what classical force an atom should be subjected to when it is in an arbitrary superposition of internal states. Spatial degrees of freedom are considered to be an environment, entanglement with which decoheres the internal states. Atoms follow classical trajectories through space, punctuated by probabilistic jumps between spin states. We modify the conventional Monte Carlo wavefunction method to jump between states when population transfer occurs, rather than when population is later discarded via exponential decay. This results in a spinor wavefunction that is continuous in time, and allows us to model the classical particle trajectories (evolution of the environment variables) more accurately. The model is not computationally demanding and it agrees well with simulations of the full spatial wavefunction of an atom.

DOI
10.48550/arxiv.1502.06674

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We present a Monte Carlo wavefunction method for semiclassically modeling spin-12\frac1221​ systems in a magnetic field gradient in one dimension. Our model resolves the conflict of determining what classical force an atom should be subjected to when it is in an arbitrary superposition of internal states. Spatial degrees of freedom are considered to be an environment, entanglement with which decoheres the internal states. Atoms follow classical trajectories through space, punctuated by probabilistic jumps between spin states. We modify the conventional Monte Carlo wavefunction method to jump between states when population transfer occurs, rather than when population is later discarded via exponential decay. This results in a spinor wavefunction that is continuous in time, and allows us to model the classical particle trajectories (evolution of the environment variables) more accurately. The model is not computationally demanding and it agrees well with simulations of the full spatial wavefunction of an atom.
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The spin coherent state path integral describing the dynamics of a spin-½ system in a magnetic field of arbitrary time dependence is considered. Defining the path integral as the limit of a Wiener regularized expression, the semiclassical approximation leads to a continuous minimal action path with jumps at the endpoints. The resulting semiclassical propagator is shown to coincide with the exact quantum mechanical propagator. A nonlinear transformation of the angle variables allows for a determination of the semiclassical path and the jumps without solving a boundary-value problem. The semiclassical spin dynamics is thus readily amenable to numerical methods.
A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach which describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin-1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wavepacket in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory or semiclassical approaches.
A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach that describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin 1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wave packet in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory, or semiclassical approaches.
Abstract Following a comprehensive analysis of the historical literature, we model the geometry of the Stern–Gerlach experiment to numerically calculate the magnetic field using the finite-element method. Using this calculated field and Monte Carlo methods, the semiclassical atomic translational dynamics are simulated to produce the well-known quantized end-pattern with matching dimensions. The finite-element method used provides the most accurate description of the Stern–Gerlach magnetic field and end-pattern in the literature, matching the historically reported values and figures.